353 replies to this topic

[Boar Wrinklestorm]

With reference to the time lag on natural processes, and with small signals getting 'lost' I've been a good doggy, and I've generalised ......

I see it more as a leaky integrator system (for each component) of the weather:




Where alpha and beta are positive constants, a is the current state, and s is the rate of change to a.

This system is called a leaky integrator because it is analagous to a bucket of water where the bucket has a hole in it, and there is a hose filling up the bucket with water. To clarify .... the rate at which the hose is filling up the bucket is s, and the rate at which water is leaking out of the bucket is directly proportional to a - the depth of water in the bucket. The constants change the system for various rates of leakage etc etc.

As always, a picture says a thousand words ... here's an excel graph modelling a hypothetical system ....



Here, s, is added to the system for a short period of time, and then withdraw to zero. at t=16. You can see clearly, given the constants, that although the system reacts quickly to input, it reacts very slowly by itself when input is withdrawn.

So, let's have a go at modelling the 11 year sun cycle using the same constants. As expected it settles down into some sort of equilibrium, which is, that all other things are held constant, the system settles down into a predictable state - all other things remaining constant have been added as an additional factor of 0.05 added to s, and denoted as c.



Now, let's try increment the constant,c, by a very small amount, say 0.01, from about t=50, for each t.



Here, you can see that the solar cycle is still evident in a rising a; has the very small increments of c modified the picture somewhat? It's not evident from the sinusoidal curves of the solar effect is it? But taken on the long term which way does the trend go?

Of course, this is just linear additions of c, when, I suspect, that CO2 are more exponential, and also, the model reacts well to reaching a higher equilibrium if you turn off c at some arbitrary late stage.

But.... the point is - at which point did you notice the solar cycles disappear? The answer is, of course, they simply didn't. Not only that but the solar cycles are still, by a factor of three, more influential on the outcome of a, than c - even though the very small value of c (1/3rd of solar) has pushed up a to almost double.

EDIT: the more observant of you will notice that I used da/dt, not dy/dt - and that is because a stands for activation of a membrane on a neural network (and I couldn't be ar5ed to change the equation)

Can I have a doggy biscuit, now, please?

Edited by VillagePlank, 16 December 2008 - 15:08 .

[Boar Wrinklestorm]

... and just for a laugh, here's what happens if we have a nuclear war, kill off the sunlight, and c emmissions reduce to 0.25.



Only one thing for it, then: where's that bunker - at the end of the flat earth, perhaps? Or, more likely, I think, to be strapped on one of those turtles, because, of course, it's turtles all the way down ....

[Gray-Wolf]

and what if 's' was able , over time ,to seal the puncture in the bucket?

[Boar Wrinklestorm]

So that a couldn't reduce?

Well it'd heat up an awful lot, then.

[Captain_Bobski]

Thundery wintry showers, on 16 Dec 2008, 03:05 PM, said:

Here's a point in a post above that I think needs addressing:

I apologise for that comment - I was exaggerating your claim (unfairly, perhaps) to make a point. Why would you expect temperatures to be "quite a bit lower" than they are now? And how much is "quite a bit"? Presumably your expectations are once again driven by the preconceived idea that solar effects are minimal, and that the lag is negligible (2 years or less).

The reason is that if solar effects were large, we would surely expect there to be a large cooling as a result of solar activity changes. In reality, we haven't seen any significant cooling (maybe 0.1-0.2C at the very most) and much of this can be explained by ENSO anyway, so the suggestion is that either recent solar activity has had a minimal effect, or something else must be offsetting the large effect that it has had.

Hi TWS,

I think you may have missed the gist of my argument, which is that Earth climate reaction to solar changes is subject to a time-lag (and, further, the suggestion that around fifty years of over-active Sun may have had a cumulative effect on Earth's climate, as I think VP was analogising).

If Earth's climate lags behind solar activity then we wouldn't expect the current solar minimum to have affected Earth climate yet. I do not see that this is a particularly off-the-wall suggestion, since several scientific studies allude to a climatological time-lag - it's the length of the lag that is at issue, I think.



CB

[Boar Wrinklestorm]

Just in case anyone thinks I making my last posts up .... here's it more formally, and, using my modified version (I use the beta constant version added as recommended in neural network stuff) here's the chart, with s increasing by 1, per t, and the constants set as displayed.



Cap'm: I am not really analagising (sp?) anything, really, just pointing out two critical points:

(i) The CO2 signal is much much smaller than any natural signal - but it is still there, and is easily seen. EDIT: a correction, there is small but cumulative temperature increase signal in the temperature record (trying to be AGW-Agnostic, here!)
(ii) Time lags aren't obvious even with a linear differential system such as the leaky integrator - which can generate all sorts of strange shapes (including sawtooth patterns) given the right set of constants

I'm not saying I'm right, and I'm not saying either +AGW or -AGW are right, I'm just saying it is mathematically possible, and models that model the real world, such as the leaky integrator - which models potential on a neuron's membrane - already use this, and no one seems to mind time lags, or small insignificant signals (vitally important in neural nets - it's what makes them resilient to error)

Edited by VillagePlank, 16 December 2008 - 17:27 .

[Captain_Bobski]

Fair points well made, VP! I read it as a mathematical analogy, but I suppose it's not really an analogy so much as a simplified mathematical model. Good work though.

Since you've got the model all worked out there, if you've got time do you think you could mock up an example with the 11-year cycle varying randomly in intensity, so we can see how the trend alters over time? You could do it twice - with and without c.

Cheers

CB

[Devonian]

VillagePlank, on 16 Dec 2008, 04:58 PM, said:

With reference to the time lag on natural processes, and with small signals getting 'lost' I've been a good doggy, and I've generalised ......

I see it more as a leaky integrator system (for each component) of the weather:

.....

Can I have a doggy biscuit, now, please?

Please, Sir, can you explain that again? . Or, god I hate maths 'let x = 7 and Y be an inverse proportional, then dx/dt = 'heck i don't know '' . Climate I get in one read, maths I read it again and again and nothing drops . Or, I kind of get it but it's like that branch just out of reach, I can see it but I just can't grab it somehow . It might be because I'm not quite sure how the text relates to the graphs (labels?). This "But.... the point is - at which point did you notice the solar cycles disappear? The answer is, of course, they simply didn't. Not only that but the solar cycles are still, by a factor of three, more influential on the outcome of a, than c - even though the very small value of c (1/3rd of solar) has pushed up a to almost double." I don't get. Which graph? The third one? Then the line wiggles due to solar but rises due to CO2? Or the wiggles are three times as big as the rate of rise?


Or, are you trying to explain how an input over time works and what happens if you change any input?

[Boar Wrinklestorm]

I presume you get chart one? Given an input of s (the bottom line) the leaky integrator gives the curve, denoted by a. This shows that you can switch off the input, s, and it takes ages to reduce in value.

Chart two shows what happens if you vary 's' rather like solar cycles - look at the blue line at the bottom, and compare to the red line at top. Because the system was started at zero it takes a little while to reach equilibrium. That is, using the bucket metaphor, there is, eventually, enough quantity of water in the bucket to, more or less, force out the same amount of water that is coming in.

Chart three shows what happens if you upset the input, in this case, the same as chart two, by increasing the input by a very small cumulative amount. It upsets the system so much that it never reaches equilibrium - at least not in this time frame. It alos shows how a very very small value can have very profound effects on a linear system (we all know it's affect on non-linear, or chaotic, systems)

Does this help?

[Devonian]

VillagePlank, on 16 Dec 2008, 08:04 PM, said:

Does this help?

I'm getting there, slowly ....

[Boar Wrinklestorm]

Devonian, on 16 Dec 2008, 07:25 PM, said:

I'm getting there, slowly ....

I've sent you the spreadsheet - let me know how you get on

[Boar Wrinklestorm]

Captain_Bobski, on 16 Dec 2008, 05:34 PM, said:

Since you've got the model all worked out there, if you've got time do you think you could mock up an example with the 11-year cycle varying randomly in intensity, so we can see how the trend alters over time? You could do it twice - with and without c.

Bit busy, today, so I've done a basic one.

The following chart uses sunspot count as it's input, s, and the constants are displayed on the chart, and it uses the leaky integrator (discussed above) to give the quantity of 'a' a 'memory' I make no theory associated with this!



EDIT: sunspot numbers are from here

EDIT2: Does anyone know where I can find the data for volcanic eruptions and magnitude of volcanic eruptions since 1700AD or similar, please?

Edited by VillagePlank, 17 December 2008 - 10:44 .

[Captain_Bobski]

That's fantastic VP - exactly what I was looking for! Thank you very much for taking the time to do that for me

Now, I'm assuming that alpha and beta are arbitrary values, so this isn't necessarily an accurate picture of the way the change in temperature is affected by the change in solar activity in the real world, but it illustrates my basic point.

Look at "s", the blue line (solar activity) circa 1956 - this is the highest peak, and yet the highest peak in "a", the temperature proxy in this argument, isn't until around 1995. By dint of our leaky integrator we find that the more recent cycles, while lower, have caused "a" to peak far later than the peak in solar activity due to the time it takes for the extra input in 1956 to "drain" out of the system.

Does anyone agree with this basic argument?



CB

[Boar Wrinklestorm]

Captain_Bobski, on 17 Dec 2008, 10:58 AM, said:

I'm assuming that alpha and beta are arbitrary values, so this isn't necessarily an accurate picture of the way the change in temperature is affected by the change in solar activity in the real world, but it illustrates my basic point.

Look at "s", the blue line (solar activity) circa 1956 - this is the highest peak, and yet the highest peak in "a", the temperature proxy in this argument, isn't until around 1995. By dint of our leaky integrator we find that the more recent cycles, while lower, have caused "a" to peak far later than the peak in solar activity due to the time it takes for the extra input in 1956 to "drain" out of the system.

Alpha and Beta are arbitrary values, but the description of how it takes time to, firstly, reach a peak, and to secondly, drain from a peak, is entirely the point of a leaky integrator. Everyone (I think) in climatology talks about how the oceans have memory and this model shows what happens when you give energy input a 'memory'

Again, it's worth saying, that this is only a model of 'memory', where, apart from the sunspot values, everything else has been set to an arbitrary value to make it easy to spot the 'memory': an important and crucial caveat.

Edited by VillagePlank, 17 December 2008 - 11:06 .

[Boar Wrinklestorm]

... and this is simply mischievous ....

[Captain_Bobski]

VillagePlank, on 17 Dec 2008, 11:06 AM, said:

Alpha and Beta are arbitrary values, but the description of how it takes time to, firstly, reach a peak, and to secondly, drain from a peak, is entirely the point of a leaky integrator. Everyone (I think) in climatology talks about how the oceans have memory and this model shows what happens when you give energy input a 'memory'

Again, it's worth saying, that this is only a model of 'memory', where, apart from the sunspot values, everything else has been set to an arbitrary value to make it easy to spot the 'memory': an important and crucial caveat.

Duly noted, VP - that's what I thought. So basically if we knew what the real world values of alpha and beta were, plus the value (static or varying) of "c" (or rather c,d,e,f...) then we might be able to cobble together a more accurate model.

As you say, though, your graph shows the principle of my argument perfectly: namely that a lack of increase in solar activity over the last 20-30 years does not necessarily negate the possibility that the Sun could be largely responsible for observed warming trends over that period.

If there is, indeed, a solar activity/climate time lag then the big cycle of the 50s, in addition to higher-than-average cycles since then, could legitimately have caused warming by a cumulative effect.

Thanks again for taking the time to plot that graph for us, VP

CB

VillagePlank, on 17 Dec 2008, 11:30 AM, said:

... and this is simply mischievous ....

misch.PNG

Nothing wrong with a bit of mischief at Christmas time, is there?

Edited by Captain_Bobski, 17 December 2008 - 11:32 .

[Boar Wrinklestorm]

Captain_Bobski, on 17 Dec 2008, 11:32 AM, said:

Nothing wrong with a bit of mischief at Christmas time, is there?

Just remember that you saw it here, first

Captain_Bobski, on 17 Dec 2008, 11:32 AM, said:

So basically if we knew what the real world values of alpha and beta were, plus the value (static or varying) of "c" (or rather c,d,e,f...) then we might be able to cobble together a more accurate model.

Yeah - I've been thinking about that .... working out the Milankovich cycles as I type, where, I presume, they will form a multiple of 's' (being that the closer to the sun, the more effect sunspots have? - I might just use sun/earth distance, actually - being that I'm far too lazy.....)

Complete conjecture, of course

[Captain_Bobski]

VillagePlank, on 17 Dec 2008, 12:30 PM, said:

Yeah - I've been thinking about that .... working out the Milankovich cycles as I type, where, I presume, they will form a multiple of 's' (being that the closer to the sun, the more effect sunspots have? - I might just use sun/earth distance, actually - being that I'm far too lazy.....)

Complete conjecture, of course

Hi VP,

that's a good place to start. I wonder also if there's any way of adding negating factors like major volcanic eruptions at the appropriate intervals (Vesuvius, Mount St Helens etc)? It would certainly be interesting to see how adding complications to the system alters the effect on the temperature proxy, "a".

Great work!



CB

[Boar Wrinklestorm]

Yeah, it's simply a subtraction from s, so that, rearranged ...

s = eb-d

where e=sunspot count, b=deviation of earth sun distance from AU, and d=volcanic disruption, and then you plug in s to the leaky integrator as before.

Also, I think you'd need to adjust s for the effect of different land masses. For instance, we might assume that the model stands for oceans (known time lag?) but for the other 1/3 you might want to reduce 'memory' of land, and eliminate memory from ice. Perhaps. I don't know.

The difficulty in all these parameters will be quantification (how much did that volcano stop incoming sunspot energy, say) and scaling (how can we scale our value to fit the model)

I think I've got the scaling, and quantification right for the earth/sun distance by using it as a multiplier of the difference to AU (the average earth sun distance) so that if the sun is further away, the mutliplier might by 0.95 (reducing s) and if it's closer it might be 1.03 (increasing s)

[Captain_Bobski]

Blimey - it's starting to get surprisingly complicated already, isn't it? (But that's a whole different argument ) You're right about quantification - I mean, we're already stumbling since we can't quantify the basic activity/climate lag (somewhere between 0 and 100 years has been mentioned, though it might be fairer to say there's a lag of 0-10 years in some parts of the system and a lag of up to 100 years in other parts of the system - complicated either way).

It might be a good place to start by assuming a basic time lag (as you have) and then quantifying other factors as fractions of the initial assumption.

Since we're just (at this point!) going for a rough analogy of the climate system, should we assume that all major volcanic eruptions block out 50% of incoming solar radiation for a period of, say, 2 years? Just to see what happens to "a".

Shall we complicate things as simplistically as possible (if that makes sense)?!

Is it also worth making changes one at a time to the equation and running a separate graph for each change? That way we can see, graphically, what a difference each factor has made to the previous run.




CB